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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic

Witold Respondek

Normandie Université, France INSA de Rouen, LMI ICMAT, Madrid, December 4, 2015

Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic

1

Introduction

2

State-space equivalence and linearization

3

Feedback equivalence and linearization

4

Orbital feedback equivalence and linearization

5

Linearization via dynamic feedback and flatness

6

4 Definitions of flatness

7

Flat systems of minimal differential weight

8

Conclusions

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Introduction

Summary

1

Introduction

2

State-space equivalence and linearization

3

Feedback equivalence and linearization

4

Orbital feedback equivalence and linearization

5

Linearization via dynamic feedback and flatness

6

4 Definitions of flatness

7

Flat systems of minimal differential weight

8

Conclusions

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Introduction

Class of control systems finite-dimensional smooth time-continous We will consider Ξ : ˙ x = F(x, u) x ∈ X, state space, an open subset of Rn u ∈ U, set of control values, a subset of Rm F is smooth (Ck or C∞) with respect to (x, u) a control system is an underdetermined system of differential equations: n equations for n + m variables

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Introduction

Very often: control-affine systems Σ : ˙ x = f(x) +

m

∑

i=1

uigi(x), x ∈ X ⊂ Rn, u ∈ Rm f and g1, . . . , gm are smooth vector fields on X state-dependent nonlinearities common in applications

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Introduction

Linearization problem

When is Ξ or Σ equivalent (transformable) to a linear control system? define equivalence (or the class of transformations) find conditions for linearization construct linearizing transformations

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization

Summary

1

Introduction

2

State-space equivalence and linearization

3

Feedback equivalence and linearization

4

Orbital feedback equivalence and linearization

5

Linearization via dynamic feedback and flatness

6

4 Definitions of flatness

7

Flat systems of minimal differential weight

8

Conclusions

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization

The system Ξ : ˙ x = F(x, u), x ∈ X, u ∈ U and ˜ Ξ : ˙ z = ˜ F(z, u), z ∈ Z, u ∈ U (the same control) are state-space equivalent, shortly S-equivalent, if there exists a diffeomorphism z = Φ(x) such that ∂Φ ∂x · F(x, u) = ˜ F(Φ(x), u) i.e., Φ∗F = ˜ F. the Jacobian matrix of Φ (the derivative of Φ, i.e, the tangent map

• f Φ) maps the dynamics F of Ξ into ˜

F of ˜ Ξ A diffeomorphism is a map Φ such that

Φ is bijective Φ and Φ−1 are Ck (C∞)

A (local) diffeomorphism defines a (local) nonlinear change of coordinates z = Φ(x)

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization

S-equivalence preserves trajectories

X x0 x(t,x0,u) Z z0 z(t,z0,u) ϕ

The image under Φ of a trajectory of Ξ is a trajectory of ˜ Ξ corresponding to the same control.

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization

S-linearization

Problem 1 When is Σ S-equivalent to a linear system, i.e., when does there exist z = Φ(x) transforming Σ into a linear system of the form ˙ z = Az +

m

∑

i=1

uibi, x ∈ Rn that is, for 1 ≤ i ≤ m, ∂Φ ∂x (x) · f = Φ∗f = Az and ∂Φ ∂x (x) · gi(x) = Φ∗gi = bi We want the same diffeomorphism Φ to transform f into Az (a linear vector field) and gi into bi, for 1 ≤ i ≤ m (constant vector fields)

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization

Why is S-linearization interesting? If we want to solve a control problem for Σ and Σ is S-equivalent to a linear system Λ, then transform Σ into Λ solve the problem for the linear system Λ transform the solution (via the inverse Φ−1 of Φ) we identify intrinsic nonlinearities

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization

A little bit of geometry: Lie bracket

Given two vector fields f and g on X, we define their Lie bracket as [f, g](x) = ∂g ∂x (x)f(x) − ∂f ∂x (x)g(x) It is a new vector field on X. It is a geometric (invariant) object Φ∗[f, g] = [Φ∗f, Φ∗g]. It measures to what extent the flows of f and g do not commute

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization

Define ad0

f g

= g adfg = [f, g] and, inductively, adk

f g

= [f, adk−1

f

g] = [f, . . . , [f, g], ] For the single-input system ˙ x = f(x) + ug(x) the Lie bracket adfg = [f, g] = [f, f + g] measures to what extent the trajectories of f (corresponding to u ≡ 0) do not commute with those of f + g (corresponding to u ≡ 1).

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization

Theorem Σ is, locally around x0, S-equivalent to a controllable linear system Λ if and only if (SL1) span {adq

f gi(x0) : 1 ≤ i ≤ m, 0 ≤ q ≤ n − 1} = Rn

(SL2) [adq

f gi, adr f gj] = 0, for 1 ≤ i, j ≤ m, 0 ≤ q, r ≤ n

Interpretation

(SL1) guarantees controllability of Λ (SL2) implies that all iterative Lie brackets containing at least two gi’s vanish, i.e., [L0, L0] = 0, where L0 is the strong accessibility Lie algebra.

Verification

(SL1) and (SL2) are verifiable in terms of f and gi’s using differentiation and algebraic operations only (no need to solve PDE’s)

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization

Theorem Σ on X is globally S-equivalent to a controllable linear system Λ on Rn if and only if (SL1) span {adq

f gi(x0) : 1 ≤ i ≤ m, 0 ≤ q ≤ n − 1} = Rn

(SL2) [adq

f gi, adr f gj] = 0, for 1 ≤ i, j ≤ m, 0 ≤ q, r ≤ n

(SL3) the vector fields f, g1,. . . ,gm are complete (equivalently, adq

f gi, 1 ≤ i ≤ m, 0 ≤ q ≤ n − 1 are complete).

(SL4) X is simply connected If we drop (SL4), then Σ is globally S-equivalent to a controllable linear system Λ on Tk × Rn−k.

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization

Constructing linearizing coordinates

Assume, for simplicity, the scalar-input case m = 1. In order to find the linearizing diffeomorphism z = Φ(x) solve the system of n 1st order PDE’s: (S) ∂Φ ∂x A(x) = Id, where A(x) = (A1(x), . . . , An(x)) and Aq(x) = adq−1

f

g(x), for 1 ≤ q ≤ n. (SL2) form the integrability conditions for (S) and assure the existence

• f solutions.

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization

Do not confuse S-linearization with linear approximation Assume F(x0, u0) = 0. The linear approximation of ˙ x = F(x, u) is ˙ z = Az + Bv + higher order terms ˙ z = Az + Bv, where A = ∂F

∂x (x0, u0) and B = ∂F ∂u (x0, u0)

So we neglect (erase) higher order terms In S-linearization higher order terms are compensated via the diffeomorphism Φ (no terms are neglected)

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization

Consider the pendulum

θ

l m g

The states are (x1, x2) = (θ, ˙ θ) and the control is the torque u

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization

The equations are ˙ x1 = x2 ˙ x2 = − g

l sin x1 + 1 ml2 u.

We have f =

• x2

− g

l sin x1

• , g =

1 ml2

• , adfg = −

1

ml2

• yielding

[g, adfg] = 0 but [adfg, ad2

f g] = 0

which implies that the pendulum is not S-linearizable

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization

But put u = ml2( g

l sin x1 + v)

we get the linear controllable system (in the Brunovsky form) ˙ x1 = x2 ˙ x2 = v. therefore there are systems that become linear after applying a (nonlinear) transformation in the control space

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Feedback equivalence and linearization

Summary

1

Introduction

2

State-space equivalence and linearization

3

Feedback equivalence and linearization

4

Orbital feedback equivalence and linearization

5

Linearization via dynamic feedback and flatness

6

4 Definitions of flatness

7

Flat systems of minimal differential weight

8

Conclusions

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Feedback equivalence and linearization

The systems Ξ : ˙ x = F(x, u), x ∈ X, u ∈ U and ˜ Ξ : ˙ z = ˜ F(z, v), z ∈ Z, v ∈ V not the same control are feedback equivalent, shortly F-equivalent, if there exists a diffeomorphism z = Φ(x) and a control transformation v = Ψ(x, u), invertible with respect to u such that ∂Φ ∂x · F(x, u) = ˜ F(Φ(x), Ψ(x, u)). Why is F-equivalence interesting?

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Feedback equivalence and linearization

Does F-equivalence preserve trajectories?

X x0 γ(t) Z φ(γ(t)) = ˜ γ(t) φ

Is the image of a trajectory, via the diffeomorphism z = Φ(x), a trajectory?

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Feedback equivalence and linearization

Yes, the image of a trajectory of Ξ, for a control u(t), is a trajectory of ˜ Ξ corresponding to v(t) = Ψ(x(t), u(t))

X x0 x(t,x0,u) Z z0 z(t,z0,v) ϕ

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Feedback equivalence and linearization

Therefore, F-equivalence preserves the set of all trajectories (the totality

• f trajectories)

X x0 Z z0 ϕ

F-equivalence is thus interesting for all problems that depend on the set

• f all trajectories (and not on a particular parametrization with respect to

control). Examples of such problems are: point-to-point controllability, trajectory tracking, stabilization.

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Feedback equivalence and linearization

Problem 2 When is Ξ F-equivalent to a linear system, i.e., when do there exist z = Φ(x) and Ψ(x, u) transforming Ξ into a linear system of the form ˙ z = Az +

m

∑

i=1

uibi, x ∈ Rn?

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Feedback equivalence and linearization

For control affine systems ˙ x = f(x) +

m

∑

i=1

uigi(x), x ∈ X we apply z = Φ(x) and control-affine feedback transformation u = α(x) + β(x)v, where the matrix β is invertible.

˙ x = f + gu u x α + βv v

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Feedback equivalence and linearization

Let D = span {f1, . . . , fk} be a distribution spanned by vector fields D is involutive if [fi, fj] ∈ D, for any 1 ≤ i, j ≤ k Put Dj = span {adq

f gi; 1 ≤ i ≤ m, 0 ≤ q ≤ j − 1}

Theorem Σ is, locally around x0, F-equivalent to a controllable linear system Λ if and only if (FL1) dim Dj(x)=const. (FL2) dim Dn(x) = n (FL3) Dj are involutive, for 0 ≤ j ≤ n (FL2) guarantees controllability of Λ (FL1)-(FL3) are verifiable in terms of f and gi’s using differentiation and algebraic operations only (no need to solve PDE’s) Geometry: D1 ⊂ · · · ⊂ Dn−1 ⊂ Dn = TX.

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Feedback equivalence and linearization

Assume, for simplicity m = 1. Involutivity of Dn−1 (of dimension n − 1 at any x) is equivalent to the existence of a family of hypersurfaces Hc = {x ∈ X : h(x) = c} tangent to Dn−1

h=c h=c' h=c''

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Feedback equivalence and linearization

Constructing linearizing transformations

The normal vector to the hypersurface Hc has to be annihilated by g, . . . , adn−2

f

g spanning Dn−1. So solve (S) ∂h ∂x A(x) = 0, where A(x) = (g(x), . . . , adn−2

f

g(x)) any solution h, dh = 0 of (S) gives linearizing coordinates zi = Li−1

f

h, for 1 ≤ i ≤ n and linearizing feedback v = Ln

f h + uLgLn−1 f

h

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Orbital feedback equivalence and linearization

Summary

1

Introduction

2

State-space equivalence and linearization

3

Feedback equivalence and linearization

4

Orbital feedback equivalence and linearization

5

Linearization via dynamic feedback and flatness

6

4 Definitions of flatness

7

Flat systems of minimal differential weight

8

Conclusions

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Orbital feedback equivalence and linearization

For the system Ξ : dx dt = ˙ x = F(x, u), x ∈ X, u ∈ U define a new time scale τ such that dt dτ = γ(x(t)), where γ is a nonvanishing function on X. With respect to the new time scale τ dx dτ = dx dt dt dτ = γ(x)F(x, u). We change the velocity along the trajectories.

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Orbital feedback equivalence and linearization

The systems Ξ : ˙ x = F(x, u), x ∈ X, u ∈ U and ˜ Ξ : ˙ z = ˜ F(z, v), z ∈ Z, v ∈ V not the same control are orbitally feedback equivalent, shortly OF-equivalent, if there exists a diffeomorphism z = Φ(x) and a control transformation v = Ψ(x, u), invertible with respect to u a nonvanishing function γ on X such that ∂Φ ∂x · γ(x)F(x, u) = ˜ F(Φ(x), Ψ(x, u)). Why is OF-equivalence interesting?

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Orbital feedback equivalence and linearization

Does OF-equivalence preserve trajectories?

X x0 γ(t) Z φ(γ(t)) = ˜ γ(t) φ

Is the image of a trajectory, via the diffeomorphism z = Φ(x), a trajectory?

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Orbital feedback equivalence and linearization

Yes, the image of a trajectory of Ξ, for a control u(t), is a trajectory of ˜ Ξ corresponding to v(t) = Ψ(x(t), u(t)) and parameterized by the new time τ = t

ds γ(x(s))

X x0 x(t,x0,u) Z z0 z(t,z0,v) ϕ

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Orbital feedback equivalence and linearization

Therefore, OF-equivalence preserves the set of all trajectories (the totality of trajectories) as unparameterized curves

X x0 Z z0 ϕ

OF-equivalence is thus interesting for all problems that depend on the set of all trajectories and not on a particular parametrization with respect to control and time.

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Orbital feedback equivalence and linearization

For Σ : ˙ x = f(x) + ∑m

i=1 uigi(x), define

the distributions G = span {g1, . . . , gm}, Gj

f

= span {f, gi, adfgi, . . . , adj−1

f

gi, 1 ≤ i ≤ m}, for 1 ≤ j ≤ n + the differential forms ωj(h) = 0, for any h ∈ Gn

f ,

ωj(adn

f gi)

= δj

i

and the functions: Tk,l

i,j = ωk([adn−1 f

gi, adl

fgj])

Attach to Σ the distribution D = span{f, g1, . . . , gm} = span{f} + G.

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Orbital feedback equivalence and linearization

Theorem (ShunJie Li-Respondek) The following conditions are equivalent: Σ is, locally around x0, OF-equivalent to a controllable linear system Λ Σ satisfies

(OFL1) dim Gn+1

f

(x) = (n + 1)m + 1; (OFL2) [Gj

f , Gj f ] ⊂ Gj+1 f

, for 1 ≤ j ≤ n; (OFL3) [G, G2

f ] ⊂ G2 f ;

(OFL4) The functions T k,l

i,j equal zero or one.

Σ satisfies

(OFL1)’ C(D(1)) = G, where C = C(D(1)) is the characteristic distribution of D(1) = [D, D], i.e., [C, D(1)] ⊂ D(1). (OFL2)’ DΣ is locally equivalent to the Jn(1, m)contact system.

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Orbital feedback equivalence and linearization

(OFL1)-(OFL4) are generalizations of involutivity conditions for feedback linearization (OFL1)-(OFL4) are verifiable in terms of f and gi’s using differentiation and algebraic operations only (no need to solve PDE’s)

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Linearization via dynamic feedback and flatness

Summary

1

Introduction

2

State-space equivalence and linearization

3

Feedback equivalence and linearization

4

Orbital feedback equivalence and linearization

5

Linearization via dynamic feedback and flatness

6

4 Definitions of flatness

7

Flat systems of minimal differential weight

8

Conclusions

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Linearization via dynamic feedback and flatness

Example: Unicycle

The unicycle on the plane subject to a nonholonomic constraint: the wheel is not allowed to slide. (x1, x2) ∈ R2: the position of the mid-point of the unicycle; θ ∈ R: the angle between the wheel and x1-axis; (u1, u2) ∈ R2: controls allowing to move (forward and backward) the unicycle and to turn. Figure: The unicycle system

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Linearization via dynamic feedback and flatness

  ˙ x1 ˙ x2 ˙ θ   = u1   cos θ sin θ   + u2   1   = u1g1 + ug2. We have [g1, g2] = Dg2 · g1 − Dg1 · g2 =   sin θ − cos θ   / ∈ span {g1, g2}. The distribution spanned by the control vector fields D = span {g1, g2} is not involutive. Thus the unicycle is not static feedback linearizable, i.e., not F-equivalent to the controllable linear system ˙ z = Az + Bv (even, locally on X ⊂ R3). But...

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Linearization via dynamic feedback and flatness

Consider the control system (precom- pensator) ˙ y = v1 and link it to the unicycle via u1 = y u2 = v2

v1 u1 v2=u2 Unicycle

Figure: The precompensated unicycle We control the derivative ˙ u1 = v1 of the first control (the second control u2 = v2 remaining the same). ⇒ Dynamic precompensation (preintegration) The precomompensated unicycle becomes F-linearizable (the linearizability distributions are involutive

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Linearization via dynamic feedback and flatness

The precomompensated unicycle becomes     ˙ x1 ˙ x2 ˙ θ ˙ y     =     y cos θ y sin θ v2 v1     applying the coordinates change     z1 z2 z3 z4     =     x1 x2 y cos θ y sin θ     and the control transformation ˜ v1 ˜ v2

• =
• cos θ

−y sin θ sin θ y cos θ v1 v2

• we get the linear controllable system

˙ z1 = z3 ˙ z2 = z4 ˙ z3 = ˜ v1 ˙ z4 = ˜ v2,

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Linearization via dynamic feedback and flatness

Remarks: It is a dynamic feedback since v1 = ˙ u1. The feedback law is invertible for y = u1 = 0. The unicycle has the same trajectories (for u1 = 0) as a linear system. Knowing z1(t) = x1(t) and z2(t) = x2(t) we can calculate all states and control via differentiation only. The dimension of the state space is not preserved. Questions: Dynamic feedback is involved: what is a dynamic invertible feedback? How to formalize equivalence via such a transformation?

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic 4 Definitions of flatness

Summary

1

Introduction

2

State-space equivalence and linearization

3

Feedback equivalence and linearization

4

Orbital feedback equivalence and linearization

5

Linearization via dynamic feedback and flatness

6

4 Definitions of flatness

7

Flat systems of minimal differential weight

8

Conclusions

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic 4 Definitions of flatness

Infinite preintegrations

Consider also its infinite prolongation Ξ∞ :                ˙ x = F(x, u0) ˙ u0 = u1 . . . ˙ ul = ul+1 . . . The system Ξ∞ is a dynamical system (differential equation and not a control system) evolving on X × U∞ = X × U × Rm × Rm × · · · . A function on X × U∞ is C∞-smooth if, locally, it depends on a finite number of variables (it is actually a function on X × Ul) and is C∞-smooth with respect to those variables. Denote the right hand side of Ξ∞ by F ∞ and put u∞ = (u0, u1, u2, · · ·).

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic 4 Definitions of flatness

Dynamic equivalence

Definition Two control systems Ξ and ˜ Ξ are D∞-equivalent if there exists a diffeomorphism χ : X × U∞ → ˜ X × U∞ such that Dχ(x, u∞) · F ∞(x, u∞) = ˜ F ∞(χ(x, u∞)). Definition (Flatness, first version) A nonlinear control system Ξ is flat if it is D∞-equivalent to a linear controllable system Λ.

• Analogous to S-equivalence of ODE’s (dynamical systems):

Dφ(x) · f(x) = ˜ f(φ(x).

• D∞-equiv. is elegant and compact but involves infinite prolongations

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic 4 Definitions of flatness

Two systems Ξ and ˜ Ξ are dynamically equivalent, shortly D-equivalent, if there exist maps Φ and Ψ mapping trajectories onto trajectories and mutually inverse on trajectories. How to formalize?

X x0 Z z0 ϕ Ψ

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic 4 Definitions of flatness

Two control systems Ξ : ˙ x = F(x, u), x ∈ X ⊂ Rn, u ∈ U ⊂ Rm and ˜ Ξ : ˙ ˜ x = ˜ F(˜ x, ˜ u), ˜ x ∈ ˜ X ⊂ R ˜

n, u ∈ ˜

U ⊂ Rm are D-equivalent if there exist two integers l and ˜ l and two pairs of maps ˜ x = φ(x, u, ˙ u, . . . , u(l)) ˜ u = ψ(x, u, ˙ u, . . . , u(l)) and (with a different numbers of derivatives) x = ˜ φ(˜ x, ˜ u, ˙ ˜ u, . . . , ˜ u(˜

l))

u = ˜ ψ(˜ x, ˜ u, ˙ ˜ u, . . . , ˜ u(˜

l))

that map trajectories into trajectories and are mutually inverse on

• trajectories. The new states and controls depend on the old states, old

controls and their derivatives and vice-versa.

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic 4 Definitions of flatness

Dynamic precompensation

Consider the control system Ξ : ˙ x = F(x, u), x ∈ X ⊂ Rn, u ∈ U ⊂ Rm together with the precompensation Π : ˙ y = G(x, y, v), y ∈ Y ⊂ Rl, v ∈ V ⊂ Rm u = ψ(x, y, v) The precompensated system becomes Ξ ◦ Π : ˙ x = F(x, ψ(x, y, v)) ˙ y = G(x, y, v).

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic 4 Definitions of flatness

Dynamic endogeneous invertible feedback

The dynamic feedback defining the precompensation is endogeneous if y = µ(x, u, . . . , u(l)), for a smooth function µ, that is, the state of the precompensator is a function of the original state, original control and its derivatives. The dynamic feedback is said invertible if the precompensated system, together with the output u = ψ(x, y, v) is input-output invertible, that is, if we can express v = ¯ ν(x, y, u, . . . , u(l)), which, in the case of an endogenous feedback, yields v = ν(x, u, . . . , u(l)).

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic 4 Definitions of flatness

Theorem (FLMR, Jakubczyk, Pomet) Given two control systems Ξ and ˜ Ξ, the following conditions are equivalent: (i) The systems are D∞-equivalent; (ii) The systems are D-equivalent; (iii) There exist two endogeneous and invertible precompensators Π for Ξ and ˜ Π for ˜ Ξ such that the precomposed systems Ξ ◦ Π and ˜ Ξ ◦ ˜ Π are S-equivalent. Definition (Flatness, second version) A nonlinear control system Ξ is flat if it is D-equivalent to a linear controllable system Λ.

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic 4 Definitions of flatness

Example

D-equivalence does not preserve the dimension of the state space. Two control systems ˙ x1 = x2 ˙ x2 = u and ˙ ˜ x1 = ˜ u are D-equivalent. Indeed, the transformations ˜ x1 = x1 ˜ u = x2 and x1 = ˜ x1 x2 = ˜ u u = ˙ ˜ u map trajectories into trajectories and are are mutually inverse on trajectories.

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic 4 Definitions of flatness

Any linear controllable system is F-equivalent to the Brunovsky canonical form: ˙ z11 = z12 ˙ zm1 = zm2 . . . · · · . . . ˙ z1ρ1−1 = z1ρ1 ˙ zmρm−1 = zmρm ˙ z1ρ1 = v1 ˙ zmρm = vm, and thus D-equivalent to the trivial system consisting of m functions z11, . . . , zm1 with no dynamics. The trajectories of that system are arbitrary evolutions of z11(t), . . . , zm1(t) subject to no constraints, so the variables are completely free. Definition (Flatness, third version) A nonlinear control system Ξ is flat if it is D-equivalent to a trivial system with no dynamics.

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic 4 Definitions of flatness

Solving trajectory generation problem via flatness

Using flatness we easily solve the constructive controllability problem: given x0 and xT , find a trajectory joining them. Assume that Ξ is D-equivalent to a controllable linear system Λ (single-input, for simplicity), which is in Brunovsky canonical form ˙ z1 = z2 . . . ˙ zn−1 = zn−2 ˙ zn = v In order to go from z0 = φ(x0) into zT = φ(xT ), choose a Cn-function ϕ(t), t ∈ [0, T], such that ϕ(0) = z10 ˙ ϕ(0) = z20 . . . ϕ(n−1)(0) = zn0 ϕ(T) = z1T ˙ ϕ(T) = z2T . . . ϕ(n−1)(T) = znT Then the control v(t) = ϕ(n)(t) steers the system from z0 into zT and the control u(t) for the original system Ξ can be computed with the help of v(t) and its derivatives (an invertible transformation!).

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic 4 Definitions of flatness

Flatness: the most popular definition

If ˜ Ξ is a system with no dynamics and m free variables are denoted ˜ u1, . . . , ˜ um, then a direct application of the third definition (D-equivalence of Ξ and ˜ Ξ), requires the existence of a map ˜ u = ψ(x, u, . . . , u(l)) such that x = ˜ φ(˜ u, . . . , ˜ u(˜

l))

u = ˜ ψ(˜ u, . . . , ˜ u(˜

l))

since ˜ x is not present. Renaming the variables ˜ ui by ϕi as well as ˜ φ and ˜ ψ by γ and δ, respectively, gives

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic 4 Definitions of flatness

Flatness: the most popular definition Ξ : ˙ x = F(x, u), x ∈ X ⊂ Rn, u ∈ U ⊂ Rm. is flat at (x0, u0, ˙ u0, . . . , u(p)

0 ) ∈ X × U × Rmp, for p ≥ −1, if there exists m

smooth functions ϕi = ϕi(x, u, ˙ u, . . . , u(p)), called flat outputs, such that x = γ(ϕ, ˙ ϕ, . . . , ϕ(s)) u = δ(ϕ, ˙ ϕ, . . . , ϕ(s)) where ϕ = (ϕ1, . . . , ϕm).

• Remark: If ϕi = ϕi(x), for 1 ≤ i ≤ m, Ξ is x-flat.

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic 4 Definitions of flatness

To memorize flatness

Consider the mechanical control system ˙ q = v ˙ v = u m To know all trajectories (q(t), v(t)) (configurations and velocities), we apply all control forces u(t) and integrate u(t) ⇒ 1 m

• u(t)dt = v(t) ⇒
• v(t)dt = q(t)

But we can look all configuration trajectories q(t) and differentiate q(t) ⇒ ˙ q(t) = v(t) ⇒ ˙ v(t) = ¨ q(t) = u(t) so q is a flat output To integrate the control system we do not have to integrate, we differentiate only

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Flat systems of minimal differential weight

Summary

1

Introduction

2

State-space equivalence and linearization

3

Feedback equivalence and linearization

4

Orbital feedback equivalence and linearization

5

Linearization via dynamic feedback and flatness

6

4 Definitions of flatness

7

Flat systems of minimal differential weight

8

Conclusions

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Flat systems of minimal differential weight

Linear systems

Theorem (i) A linear control system Λ : ˙ x = Ax + Bu is flat if and only if it is controllable. (ii) Flat outputs are ϕ = z11, . . . , ϕm = zm1, the top variables of the Brunovsky canonical form. For m = 1, define c = 0 by cb = cAb = · · · = cAn−2b = 0. Then h = cx is a flat output

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Flat systems of minimal differential weight

Nonlinear single-input systems, m = 1

Theorem The following conditions are equivalent for a single-input system Σ (i) Σ is flat; (ii) Σ is F-linearizable (iii) Σ satisfies (FL1) dim Dj(x)=const. (FL2) dim Dn(x) = n (FL3) Dj are involutive, for 0 ≤ j ≤ n (Dn−1 is involutive) Moreover, a flat output is any function ϕ satisfying (S) ∂ϕ ∂x A(x) = 0, where L(x) = (g(x), . . . , adn−2

f

g(x))

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Flat systems of minimal differential weight

F-linearizable systems

For multi-input systems m ≥ 2, F-linearizability is sufficient for flatness but not necessary. Moreover, flat ouputs are the top variables of the Brunovsky canonical form.

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Flat systems of minimal differential weight

Minimal flat outputs and differential weight For any flat output ϕ of Ξ there exist integers s1, . . . , sm such that x = γ(ϕ1, ˙ ϕ1, . . . , ϕ(s1)

1

, . . . , ϕm, ˙ ϕm, . . . , ϕ(sm)

m

) u = δ(ϕ1, ˙ ϕ1, . . . , ϕ(s1)

1

, . . . , ϕm, ˙ ϕm, . . . , ϕ(sm)

m

). We can choose (s1, . . . , sm) such that if for any other m-tuple (˜ s1, . . . , ˜ sm) x = ˜ γ(ϕ1, ˙ ϕ1, . . . , ϕ(˜

s1) 1

, . . . , ϕm, ˙ ϕm, . . . , ϕ(˜

sm) m

) u = ˜ δ(ϕ1, ˙ ϕ1, . . . , ϕ(˜

s1) 1

, . . . , ϕm, ˙ ϕm, . . . , ϕ(˜

sm) m

), then si ≤ ˜ si, for 1 ≤ i ≤ m.

• Differential weight of ϕ = ∑m

i=1(si + 1) = ∑m i=1 si + m,

i.e., minimal number of derivatives of ϕi needed to express x and u.

• ϕ: minimal flat output if its differential weight is the lowest among all flat
• utputs of Ξ.
• Differential weight of Ξ: the differential weight of a minimal flat output.

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Flat systems of minimal differential weight

Static feedback linearizable (F-linearizable) systems F-linearizable systems are the only flat systems of differential weight n + m.

• The representation of x and u uses the minimal possible, which is n + m,

number of time-derivatives of ϕi.

• For any flat system, that is not F-linearizable, the differential weight

is bigger than n + m. measures the smallest possible dimension of a precompensator linearizing dynamically the system.

• In general, a flat system is not F-linearizable, except the single-input case

where flatness reduces to F-linearization.

• The simplest flat systems that are not F-linearizable are systems that

become F-linearizable via one-dimensional precompensator ⇒ differential weight n + m + 1.

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Flat systems of minimal differential weight

Our goal

• To give a geometric characterization of control-affine systems

Σ : ˙ x = f(x) +

m

∑

i=1

uigi(x), that become F-linearizable after a one-fold prolongation of a suitably chosen control (the simplest dynamic feedback). verifiable conditions (like involutivity conditions). describe and understand the geometry of this class of systems.

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Flat systems of minimal differential weight

Main theorem (Nicolau - Respondek) m=2 Assume k ≥ 1 and ¯ Dk = TX. Σ is x-flat at x0 ∈ X, of differential weight n + 3, if and only if (A1) rk ¯ Dk = 2k + 3; (A2) rk ( ¯ Dk + [f, Dk]) = 2k + 4 ⇒ ∃ gc ∈ D0 such that adk+1

f

gc ∈ ¯ Dk; (A3) Bi, for i ≥ k, is involutive, where Bk = Dk−1 + span {adk

f gc} and

Bi+1 = Bi + [f, Bi]; (A4) There exists ρ such that Bρ = TX.

Comparison with the F-linearizable case

• Geometry of flat systems of differential weight n + 3

D0 ⊂

2 · · · ⊂ 2 Dk−1

⊂

2

Dk ⊂

1

¯ Dk 1∪ = Bk ⊂

2

Bk+1 ⊂

2 · · · ⊂ 2 Bµ ⊂ 1 Bµ+1 ⊂ 1 · · · ⊂ 1 Bρ = TX

• Geometry of F-linearizable systems

D0 ⊂ D1 ⊂ · · · ⊂ Dn−1 = TX

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Flat systems of minimal differential weight

Remarks

• General result (the particular cases k = 0 and ¯

Dk = TX have slightly different geometry).

• Enables us to define, up to a multiplicative function, the control to be

prolonged: gc = β1g1 + β2g2 ∈ D0 ⇒ v1 = d dt (β2u1 − β1u2) = d dt ˜ u1

• All conditions are verifiable, verification involves derivations and algebraic
• perations only (without solving PDE’S).

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Flat systems of minimal differential weight

Calculating flat outputs

• µ: the largest integer such that corank (Bµ−1 ⊂ Bµ) = 2.
• ρ: the smallest integer such that Bρ = TX.

Proposition (Nicolau - Respondek) (i) Assume ¯ Dk = TX or ¯ Dk = TX and [Dk−1, Dk] ⊂ Dk. (ϕ1, ϕ2) is a minimal x-flat output at x0 if and only if d ϕ1 ⊥ Bρ−1 d ϕ2 ⊥ Bµ−1, d ϕ2 ∧ d ϕ1 ∧ dLf ϕ1 ∧ · · · ∧ dLρ−µ

f

ϕ1(x0) = 0. The pair (ϕ1, ϕ2) is unique, up to a diffeomorphism. (ii) Assume ¯ Dk = TX and [Dk−1, Dk] ⊂ Dk. (ϕ1, ϕ2) is a minimal x-flat output at x0 if and only if (d ϕ1 ∧ d ϕ2)(x0) = 0 and the involutive distribution L = (span {d ϕ1, d ϕ2})⊥ satisfies Dk−1 ⊂ L ⊂ Dk.

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Flat systems of minimal differential weight

Remarks for the case (ii) ¯ Dk = TM and [Dk−1, Dk] ⊂ Dk

• For any function ϕ1, satisfying

d ϕ1 ⊥ Dk−1, there exists ϕ2 such that the pair (ϕ1, ϕ2) is a minimal x-flat output and the choice of ϕ2 is unique, up to a diffeomorphism.

• There is as many flat outputs as functions of three variables (since Dk−1 is

involutive and of corank 3).

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Flat systems of minimal differential weight

Induction motor - first model with θ, the mechanical position

                   ˙ θ = ω ˙ ω = µψdiq − τL

J

˙ ψd = −ηψd + ηMid ˙ ρ = npω + ηMiq

ψd

˙ id = −γid + ηMψd

σLRLS + npωiq + ηMi2

q

ψd

+ ud

σLS

˙ iq = −γiq − Mnpωψd

σLRLS − npωid − ηMidiq ψd

+ uq

σLS

• ud, uq are the inputs (the stator voltages);
• id and iq are the stator currents;
• ψd and ρ are two well-chosen functions of the rotor fluxes;
• ω is the rotor speed;
• θ is the mechanical position.

The system is flat of differential weight 9 = 6 + 2 + 1 = n + m + 1. k = 1 and ¯ D1 = TX

Prop.2(i)

= ⇒ (ϕ1, ϕ2) = (ω, ρ) and the pair (ϕ1, ϕ2) is unique.

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Flat systems of minimal differential weight

Induction motor - second model without θ, the mechanical position                ˙ ω = µψdiq − τL

J

˙ ψd = −ηψd + ηMid ˙ ρ = npω + ηMiq

ψd

˙ id = −γid + ηMψd

σLRLS + npωiq + ηMi2

q

ψd + ud σLS

˙ iq = −γiq − Mnpωψd

σLRLS − npωid − ηMidiq ψd

+ uq

σLS

The system is flat of differential weight 8 = 5 + 2 + 1 = n + m + 1. k = 1, ¯ D1 = TX and [D0, D1] ⊂ D1 Prop.2(ii) = ⇒ many flat outputs (the choice being parameterized by a function of three well defined variables) if ϕ1 = ω, then ϕ2 = ρ; if ϕ1 = ψd, then ϕ2 = ηM

µψ2

d ω − ρ;

if ϕ1 = ρ + ηM

µψd , then ϕ2 = ψd − ω.

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Flat systems of minimal differential weight

Main theorem (Nicolau - Respondek) m ≥ 3 Assume k ≥ 1 and cork (Dk ⊂ [Dk, Dk]) ≥ 2. A control system Σ is x-flat, with the differential weight n + m + 1, if and only if it satisfies around: (A1) There exists an involutive subdistribution Hk ⊂ Dk, of corank one; (A2) Hi, for i ≥ k + 1, is involutive, where Hi = Hi−1 + [f, Hi−1]; (A3) There exists ρ such that Hρ = TX. Comparison with the F-linearizable case

• Geometry of flat systems of differential weight n + m + 1

D0 ⊂ · · · ⊂ Dk−1 ⊂ Dk ⊂ ¯ Dk 1∪ ∩ Hk ⊂ Hk+1 ⊂ · · · ⊂ Hρ = TX

• Geometry of F-linearizable systems

D0 ⊂ D1 ⊂ · · · ⊂ Dk−1 ⊂ Dk ⊂ Dk+1 ⊂ · · · ⊂ Dn−1 = TX

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Flat systems of minimal differential weight

Remarks

1

General result (the particular cases k = 0 and cork (Dk ⊂ [Dk, Dk]) = 1 have slightly different geometry). If k = 0: similar result, but in the chain of subdistributions H0 ⊂ D0 ⊂ H1 ⊂ H2 ⊂ · · · the distribution H1 is not defined as Hk+1 = Hk + [f, Hk], but as H1 = D0 + [D0, D0] + [f, H0] and satisfies an additional nonsingularity condition ⇒ singularity in the control space.

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Flat systems of minimal differential weight

Remarks

2

In order to verify conditions (A1)-(A3): check the existence of the involutive subdistribution Hk of corank one in Dk. Pasillas-Lépine and Respondek (2001): checkable conditions, based on Bryant (Ph.D. thesis, 1979), to verify the existence of an involutive subdistribution of corank one and an explicit way to construct it. If cork (Dk ⊂ [Dk, Dk]) ≥ 2: the subdistribution Hk is unique. If cork (Dk ⊂ [Dk, Dk]) = 1: the subdistribution Hk is no longer unique, but we can uniquely identify it by another argument.

3

All conditions are verifiable, verification involves derivations and algebraic

• perations only (without solving PDE’S).

4

Explicit construction of Hk enables us to define the control to be prolonged (given up to a multiplicative function).

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Conclusions

Summary

1

Introduction

2

State-space equivalence and linearization

3

Feedback equivalence and linearization

4

Orbital feedback equivalence and linearization

5

Linearization via dynamic feedback and flatness

6

4 Definitions of flatness

7

Flat systems of minimal differential weight

8

Conclusions

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Conclusions

What do we know about flatness?

Via flatness we can solve the constructive controllability problem Although very useful, flatness is a highly non generic property: a slight perturbation of a flat system yields a non flat one (Tcho´ n) We know that a few classes of control systems are flat: accessible systems with n − 1 controls, accessible control-linear systems with n − 1 and n − 2 controls We know to characterize flat control systems of special forms: feedback linearizable systems, control-linear systems with 2 controls (chained form), m-chained form

• r of very special dimensions: 3 states and 2 controls (nonlinear)

and 4 states and 2 controls (affine) We know to characterize flat systems of differential weight n + m + 1

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Conclusions

What don’t we know about flatness?

We do not know to characterize flatness in general. We do not know whether the problem is finite or infinite dimensional, that is, we do not know if there is a bound on the number of derivatives of controls We do not even know how to check flatness for control-affine systems with 2 controls nor for control-linear systems with 3 controls We know that the problem is difficult: Ellie Cartan (1914) has introduced the notion of absolute equivalence of underdetermined differential equations. His absolutely trivial equations are just flat systems. He proved that systems with 2 controls are flat (absolutely trivial) if and only if they are equivalent to the chained form (Goursat normal form). Cartan claimed that the general problem is difficult. Non flat systems exist! The first example is due David Hilbert (1912) who had also been working on absolute equivalence (integrating differential equations without integration). His example is, geometrically, the same as the unicycle towing a trailer but with a hook that is not at the mid-point.

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Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Conclusions

Conclusions

1

We presented various definitions of the notion of flatness

2

We provided geometric tools convenient (needed) to study flatness

3

We presented geometric conditions for characterizing flatness (verifiable via differentiation and algebraic operations only) for a few classes of systems

4

Do not confuse linearization (static, dynamic) with linear approximation

5

Whenever we can linearize the system (statically, dynamically), the control problems, we are dealing with, get substantially simplified

6

Even if we do not apply linearizing transformations or the system is not linearizable (flat), our knowledge about the system is deeper: we identify intrinsic nonlinearities that cannot be removed via feedback (static, dynamic)

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